WARPLAB

What am I looking at?

What you are looking at

A map of countries, each with its own currency, with an exchange rate on every route between neighbours. A rate on its own means nothing, because each country picks its own unit; what you can measure is what a trip round a block gains. That gain is Maldacena's picture of the electromagnetic field, and below the map gold prices add his picture of the Higgs field, which gives the field a mass.

Controls

On this device:

Mouse

  • Click a route to see its rate, then drag Nudge this rate up to 25% either way.
  • Click a block to follow a trip round it with 100 units; click a country, then Remove a zero or Add a zero.
  • Find the best loop picks the block that pays most; Change every currency changes every country's unit at random.
  • Calm world and Random world deal a new map; 2 × 2, 3 × 3 and 4 × 4 set how many countries per side.
  • Sell gold in every country puts a gold price on each country and a gold-trip gain on each route; Make gold cost 1 everywhere then turns every unit into an ounce.
  • In the essay's steps, click a step or Back and Next; drag the two sliders in step 3, and click PES, AUS or FLO in step 4 to remove a zero.

Keyboard

  • The map is one Tab stop: the arrow keys move between countries, routes and blocks, Home goes to the top-left country and End to the bottom-right one.
  • Enter or Space picks the country, route or block in focus; its details open beside the map.
  • Tab on to Nudge this rate; the arrow keys move it 0.5%, Home and End jump to −25% and +25%.
  • Every other button takes Tab, then Enter or Space.
  • In step 3, the arrow keys move the wave length by one country and the gold weight by 0.01.
  • Enter on From the economy to physics opens the dictionary.
  • H or ? opens this panel. Esc closes it.
Other devices

Touch

  • Tap a route to see its rate, then slide Nudge this rate.
  • Tap a block to follow a trip round it, or a country to remove or add a zero.
  • Tap Find the best loop, Change every currency, Calm world or Random world under the readouts.
  • Tap 2 × 2, 3 × 3 or 4 × 4 to change the map's size; a smaller map is easier to read on a phone.
  • Tap Sell gold in every country, then Make gold cost 1 everywhere.
  • Tap a numbered step or Back and Next to walk through the essay's examples.

Glossary

Map (lattice)
The countries sit on a square grid, 2, 3 or 4 per side. Neighbours are joined by routes, and every square of four countries is a block: a 3 × 3 map has 12 routes and 4 blocks.
Why it matters Physicists cut space into a grid like this to calculate fields on a computer.
Country
A dot on the map with a three-letter code, such as PES for the peso. In the dictionary below the map, a country is a point in space.
Currency
Each country's own money and its own choice of what one unit is worth. Physicists call such a local choice of reference a gauge.
Why it matters Because each country can rename its unit, no single rate is a fact about the world.
Route
The line between two neighbouring countries, where you can change one currency for the other. Its number is the exchange rate.
Exchange rate
How many units of the right (or lower) country's currency one unit of the left (or upper) country's buys. Very small or large rates are written short: 1.3E-3 means 0.0013.
Why it matters In the physics, the rates are the gauge field: the electromagnetic potential.
Nudge this rate
A slider that moves the picked route's rate up to 25% up or down. The two blocks beside the route change: one starts to pay clockwise, the other anticlockwise.
Block
A square of four countries and the four routes round it. Each block shows what one trip round it gains, with ↻ or ↺ for the way that pays.
Trip round a block
Leave with 100 units, change money at each of the four corners and come back to where you started. The ledger lists each step and what you hold after it.
Why it matters Coming back to the same currency is what makes the result comparable, whatever the units.
Loop gain
How much richer a trip round a block leaves you, in percent, going the way that pays. A gain like this, made only by changing money, is arbitrage; below 0.05% the module counts it as none.
Why it matters The loop gain is the field strength, the magnetic field: the part no convention can change.
F, the log gain round a block
Take the natural log of each rate; F is the top and right ones minus the bottom and left ones. A trip clockwise multiplies your money by e^F, so the gain is e^|F| − 1: F = 0.1 is about 10.5%. F above 0 pays clockwise, below 0 anticlockwise.
Best loop pays, Loops that pay
Best loop pays is the largest loop gain on the map. Loops that pay counts the blocks with any gain, out of all blocks; Find the best loop opens the best one.
Trader
The dot that circles each block that pays, the way that pays, faster the more it pays. The still version leaves it out.
Why it matters In the physics it is a charged particle, like an electron, pushed round by a magnetic field.
Change of currency (gauge transformation)
Every country picks a new unit, worth up to ten times more or less than the old one. The rates move; the counts below show how many changed, while the loop gains that changed stay at 0.
Why it matters Rates are convention; what a loop gains is what you can measure.
Remove a zero
One country makes a new unit worth 10 old ones. Every rate out of it grows tenfold and every rate into it shrinks tenfold; ×10 or ÷10 on the country, and Zeros removed, keep count. Add a zero undoes it.
Why it matters It is the simplest change of currency: one country, and still no loop gain moves.
Calm world (pure gauge)
A world whose rates are not 1 but where no block and no closed trip of any length gains anything. It is what you get from rates of 1 everywhere after a change of currency.
Why it matters It shows that rates far from 1 can still hide no field at all.
Random world
A calm world with every rate pushed up or down by up to about 13%. Most blocks then pay a little, some more than others.
Gold price (Au), the Higgs field
Each country also sells gold at its own price in its own money, shown as Au on the country. The module sets the prices from the rates, so that gold trips gain as little as the rates allow, with the top-left country at 40.
Why it matters In Maldacena's picture this price is the Higgs field.
Gold trip
Buy an ounce of gold on one side of a route, sell it on the other and change the money back: one route, no block. The percentage under each route's rate is what it gains; Best gold route pays is the largest.
Why it matters Once gold is sold, a single route can pay, so the rates are no longer free.
Gold at 1 everywhere (unitary gauge)
Each country takes one ounce of gold as its unit, as in the essay: a gold peso of 40 pesos, a gold dollar of 5 dollars. The rates move, no gold trip or loop does, and any rate other than 1 pays on its own route.
Why it matters In these units the gold-trip gain is plain to read off each rate.
Energy of a wave of rates
Step 3 takes a wave of rates, rising and falling across the countries, and adds up what it pays. Energy from loops falls towards 0 as the wave gets longer; energy from gold is the same at every length. Total with gold is the two together.
Why it matters How energy depends on wave length tells physicists whether a particle has a mass.
How much gold weighs in the economy
A slider from 0 to 0.8 for how much gold trips count against loops. The energy from gold is this number squared, the dashed floor on the chart.
Mass
Without gold a long wave costs almost nothing: a particle with no mass, like the photon. With gold no wave costs less than the floor: a particle with mass, like the W and Z bosons of the weak force.
Why it matters This is the job the Higgs field does in nature.
Gold / silver (the Higgs boson)
How many times dearer gold is than silver in one country. Gold at 2000 pesos and silver at 1000 make it 2; in australes of 1000 pesos they cost 2 and 1, and it is still 2.
Why it matters A quantity the same in every currency is a physical field, and its ripples are the Higgs boson found at the LHC in 2012.

All help and questions

Currencies and the Higgs

Countries trade currencies on a small map. Find the loop of exchange rates that pays, then sell gold and the field gets a mass: Maldacena's Higgs.

Each number on a route is how many units of the right (or lower) country's currency one unit of the left (or upper) one buys. Each block shows what a trip round it gains, and which way round pays.

Countries per side3 × 3

Tap a route to change its rate, a block to go round it with 100 units, or a country to remove a zero from its currency.

Best loop pays
19.3%
Loops that pay
4 of 4
Zeros removed
0

The Higgs field

Each country can also set its own price of gold. That price is the Higgs field, and it opens a trip that uses a single route.

The essay's examples, step by step

One route pays

Maldacena's example. 4 pesos buy a dollar. Gold costs 40 pesos in Argentina and 5 dollars in the United States.

  1. Leave with 5 USD
  2. → buy 1 oz of gold in the United States
  3. → sell it in Argentina for 40 ARS
  4. → change to 10 USD

Back with 10: 100% gained, on one route, with no block to go round.

Without gold, only a loop could pay. With gold, a single route does.

Step 1 of 4
From the economy to physics
  • A country · a point in space
  • Its currency · a local choice of reference: a gauge
  • The rate between neighbours · the gauge field, the electromagnetic potential
  • Removing a zero · a gauge transformation: the rates change, the physics does not
  • The gain round a loop · the field strength, the magnetic field
  • A trader who goes round the loop that pays · a charged particle, like the electron
  • The price of gold in each country · the Higgs field
  • Gold at 1 everywhere: any rate other than 1 pays on one route · the W and Z bosons get a mass
  • How many times dearer gold is than silver · the Higgs boson

The rates are Maldacena's picture of electromagnetism. Gold, below and in the steps, is his picture of the Higgs: switch it on and the routes stop being free.

Credits

Based on Juan Maldacena, “The symmetry and simplicity of the laws of physics and the Higgs boson”, arXiv:1410.6753 (European Journal of Physics).

The examples here are his, retold in our words. K. Young drew the analogy between currency markets and lattice gauge theory earlier (American Journal of Physics, 1999); Maldacena takes it on to the Higgs. WarpLab has no affiliation with either author.

Sources

Analytic model. This module uses no instrument data.

Physics benchmarks

  • Under random changes of currency in every country, every rate moves and the product of rates round each block (and round any closed loop) does not change.
  • With zero field strength on every block, no closed loop of any length gains anything, even though the rates are not 1.
  • Under random changes of currency, gold and silver prices move in every country and the gold/silver ratio does not; nor does what the gold trip gains on any route.
  • The essay's examples: with 4 pesos to the dollar and gold at 40 pesos and 5 dollars, 5 dollars come back as 10 (100%); with gold at 1 in both currencies, 1 gold peso buys 2 gold dollars and the gain stays at 100%.
  • Without gold a long wave of rates costs almost no energy; with gold none costs less than a floor (the mass), and the curve matches the wave's energy on the lattice.

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